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 A000408 Numbers that are the sum of three nonzero squares. 63
 3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS From Jonathan Vos Post, Mar 03 2010: (Start) Catalan conjectured that three times any odd square not divisible by 5 is a sum of squares of three primes other than 2 and 3 (regarding 1 as a prime). Catalan stated and Realis proved that every power of 3 is a sum of three squares relatively prime to 3. [See Dickson] From the abstract of the Berkovich and Jagy paper: "Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n) - ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. To prove this identity we employ the Siegel-Weil and the Smith-Minkowski product formulas." (End) a(n) not equal 7 mod 8. - Boris Putievskiy, May 05 2013 A025427(a(n)) > 0. - Reinhard Zumkeller, Feb 26 2015 According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - Jeffrey Shallit, Jan 15 2017 REFERENCES L. E. Dickson, History of the Theory of Numbers, vol. II:  Diophantine Analysis, Dover, 2005, p. 267. Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 87-88. (See also p. 73 where the question is posed.) Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), 11-20. LINKS Ray Chandler, Table of n, a(n) for n = 1..10000 Alexander Berkovich, Will Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2, arXiv:1101.2951 [math.NT], 2011. B. Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), J. Int. Seq. 16 (2013) #13.6.4 S. Mezroui, A. Azizi, M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8 FORMULA a(n) = 6n/5 + O(x/sqrt(log n)). (Can the error term be improved?) - Charles R Greathouse IV, Mar 14 2014 MAPLE N:= 1000: # to get all terms <= N S:= series((JacobiTheta3(0, q)-1)^3, q, 1001): select(t -> coeff(S, q, t)>0, [\$1..N]); # Robert Israel, Jan 14 2016 MATHEMATICA f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], _?Positive]]; f (* Ray Chandler, Dec 06 2006 *) pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range, pr[#] != {} &] (* Jean-François Alcover, Apr 04 2013 *) max = 1000; s = (EllipticTheta[3, 0, q] - 1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* Jean-François Alcover, Feb 01 2016, after Robert Israel *) PROG (PARI) is(n)=for(x=sqrtint((n-1)\3)+1, sqrtint(n-2), for(y=1, sqrtint(n-x^2-1), if(issquare(n-x^2-y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013 (PARI) is(n)= my(a, b) ; a=1 ; while(a^2+1 0) . a025427) [1..] -- Reinhard Zumkeller, Feb 26 2015 CROSSREFS Cf. A004214 (complement), A024795 (numbers with multiplicity), A000404, A000378, A025427. Sequence in context: A329511 A065940 A024795 * A025321 A153238 A230193 Adjacent sequences:  A000405 A000406 A000407 * A000409 A000410 A000411 KEYWORD nonn AUTHOR STATUS approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)